"Magic" diamonds for fuel?

Question

The subterranean nation of Glot has developed the ability to “thermally link” two equally sized diamonds, such that they are always the same temperature, regardless of spatial separation. We can consider this “thermal linking” process “magic” (or sci-fi "thermal quantum entanglement"), but physics otherwise operate the same as in our universe. Contrary to this particular advancement, Glot has NOT discovered electricity. Otherwise, it has technology equivalent to the early-mid 1800's.

One of the diamonds is carefully immersed in one of their many magma–rivers, the other is put into a steam engine boiler. (These magma-rivers are part of a very large underground area, and are accessible, with precautions, at minimal cost.) Always at equal temperatures, the two diamonds boil/heat the steam, and “cool” the magma river respectively.

The “linking” process is far less expensive than the diamonds themselves, so they are cut into slices, half a millimeter in thickness to increase surface area. These slices are the smallest they can mount, without having them “wash” away in the magma river, while keeping the diamond in contact with the magma.

Without considering the cost of the diamonds: would this process actually work to produce a useful, geothermal powered, steam engine? (That doesn’t require shoveling, or even carrying fuel?)

If so, what mass of diamonds, in total, would be required to power an average steam locomotive of the 19th century?

Answer 1

The convection heat transfer coefficient for water heated by a solid surface, in a typical boiler, is about 1 kilowatt per square meter and kelvin. (That's why in a fire-heated boiler you have all those pipes, through which hot combustion gas flows; it it to maximize the surface area where the hot gas transfers heat to the water.)

This means that a shard of diamond with an area of 5×5 cm = 25 square centimeters (0.0025 square meters), 1000 K hotter than the water, will be able to transfer 2x2.5 = 5 kW. (I am assuming that the diamond is placed vertically so that both surfaces can transfer heat.) Locomotive steam engines have an efficiency of about 5% tops, so overall the engine will produce about 250 W, or 0.4 horsepower. A more-or-less typical steam locomotive produces about 2000 horsepower; it results that you need some 12.5 square meters of lava-temperature diamonds for this. (And you cannot have one big diamond, or else the heat transfer coefficient will come down abruptly; you need many small ones.)

The density of diamond is 3.5 grams per cubic centimeter. Assuming that the many small diamond pieces are 1 mm thick, the 12.5 square meters of heating come to 44 kg, or 220,000 carats. For uncut large diamonds, a price of 2,000 USD per carat is reasonable, so the 220,000 carats needed for one lousy steam locomotive would cost some 440 million USD. OK, maybe in your world De Beers doesn't exist so diamond prices are not massively inflated, but we are still speaking of something like 100 million USD.

Note 1: For the curious, I started with a 5x5 cm diamond because in my mind that's about the upper limit for an enormous diamond.

Note 2: I have no idea whether it is even possible to cut diamonds in thin flat slices. Since the calculations are crude approximations anyway, this shouldn't matter much.

Note 3: Lava is rarely hotter than 900° C, so a more natural temperature differential would have been 800 kelvin. I've taken 1000 kelvin for ease of calculation.

Answer 2

Of course it would work. There's no practical difference between your diamonds and an electric heating element — except the diamond is experiencing 1,000℃ and the heating element usually isn't.

The issues are two-fold:

Surface Area: you need to get the heat off the diamond: more surface area = faster transfer.

Suspension: It would also be best to figure out how to suspend the diamond. Something that hot would eventually compromise the steel/iron/metal container it's sitting against (if it were simply thrown in), so your people would likely be experts at ceramics.